Structural Engineering & Statics
where the load goes
Concept

Damping ratio — where it appears

The damping of a vibrating system as a share of the amount that would just stop it oscillating, written ζ. It decides how high a resonance stands and how fast a free vibration dies, and at a foundation most of it is energy carried away by the ground rather than lost in any material.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

How much a harmonic force is magnified, at four damping ratios. Displacement amplitude divided by the deflection the same force would produce if it were applied slowly, against the ratio of the forcing frequency to the structure's own, at 1%, 2%, 5%, 10% of critical damping. At the natural frequency the magnification is 50, 25, 10, 5 respectively — one over twice the damping ratio, and nothing else in the problem enters it.

The only thing that stops it

Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built.

dynamics · Damping The vertical response on layers of different depth. The vertical amplitude of a 150 tonne block 5.0 m across on soil with a shear wave speed of 178 m/s, under a harmonic force of fixed size, as a multiple of its static deflection on a half-space, against frequency. On a half-space the mode is at 12.5 Hz and the peak is 1.10. On 10.0 m the mode is at 14.3 Hz against a cut-off of 7.4 and the peak is 1.03. On 6.0 m the mode is at 15.5 Hz against a cut-off of 12.3 and the peak is 1.74. On 4.5 m the mode is at 16.3 Hz against a cut-off of 16.4 and the peak is 5.84. The layer stiffens the footing and lifts the frequency a little; what changes the picture is the cut-off, which rises as the layer thins and, once it passes the mode, takes the radiation damping away and leaves only the soil's own 5 per cent. Radiation below the cut-off is taken as nothing and its rise above it as linear, which is the usual idealisation of the exact layered solution.

The rock that sends the waves back

A machine block on the ground is damped by the waves it launches, and a half-space lets them all escape. Put rock four and a half metres down and none of them can: below the soil layer's own lowest frequency there is no wave that travels, and a vertical mode damped at 47 per cent of critical keeps the soil's own 5. Its resonance grows fivefold, and the heavy block the textbook rule recommends is the one that loses its damping on the deepest ground.

dynamics · Vibration isolation Two blows on three foundations. The movement of a 150 tonne block struck by a hammer delivering 18 kN·s, twice, 0.75 s apart. On soil at 12.5 Hz with the 47 per cent of damping a half-space radiates, the block moves 0.86 mm and is still before the second blow. On the same soil over shallow rock, with 5 per cent, it moves 1.42 mm and rings for the whole interval. On springs at 4.0 Hz it moves 4.42 mm, and its ringing has not died when the second blow arrives. The peak force passed to the ground is 1.34, 1.32 and 0.42 MN: the heavily damped block and the lightly damped one pass almost the same force.

The blow that has no frequency

A forge hammer does not shake its foundation; it strikes it. The transmissibility curve every isolation design is drawn on has nothing to say about a blow, and the rules it teaches mislead. The force a struck block passes to the ground is least at a quarter of critical damping, not the most; five per cent and forty-seven pass almost the same; and the springs that soften every blow can bring the block down to the hammer's own rate and make a train of blows ring four times as high as one.

dynamics · Vibration isolation

Named alongside it

The objects these essays reach for when they reach for this one.

Natural frequencyResonanceDampingDynamic amplificationEnergy dissipationFree vibrationHysteresisImpulseLogarithmic decrementRadiation dampingServiceabilitySite response

All concepts